By [1]}, \(\color {black}\Gamma ^r\)  (and hence \(\color {black}[\Gamma ^{\prime }]^r\) ) has a approximate tangent line at \(\mathcal {H}^1\) -a.e. \(x,\)  therefore, (REF ) follows from [2]}. To prove (REF ) with \(\Gamma \) 
choose \(x\in \Gamma \)  such that \(\theta (\Gamma ,x)=1\)  and \(T_x\)  exists. Without loss of generality we assume that \(x=0\)  and \(\nu _\Gamma (x)={\bf e_2}\)  is the unit normal to \(T_x.\) 
First we prove
\(\sigma _{0,r}(\Gamma ) \overset{K}{\rightarrow } T_0\) 
as \(r\searrow 0.\)  Indeed, let \(\mu _r:=\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\sigma _{0,r}(\Gamma ))\)  and \(\mu _0:=\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits T_0.\)  Given \(r>0,\)  since \(\mu _r( Q_\rho (x))=\frac{\mathcal {H}^1( Q_{\rho r}(rx))}{r},\)  by (REF ) for all \(x\in \sigma _{0,r}(\Gamma )\)  and \(\rho \in (0,r_0/r)\)  one has
\(c\le \frac{\mu _r( Q_\rho (x))}{2\rho } \le C.\) 
Let \(r_k\searrow 0\)  be any sequence. By compactness of sets in the Kuratowski convergence, passing to a further not relabelled subsequence if necessary we suppose that
\( \sigma _{0,r_k}(\Gamma ) \overset{K}{\rightarrow } L\) 
for some closed set \(L\subset \mathbb {R}^2\)  as \(k\rightarrow \infty .\)  We claim that \(L=T_0.\)  If there exists \(x\in T_0\setminus L,\)  then for some \(\rho >0,\)  \( Q_\rho (x)\cap L=\emptyset .\) 
By (REF ), \( Q_{\rho /2}(x)\cap \sigma _{0,r_k}(\Gamma )=\emptyset \)  for all large \(k\)  so that \(\mu _{r_k}( Q_{\rho /2}(x))=0.\)  Then by (REF )
\(0=\lim \limits _{k\rightarrow \infty } \mu _{r_k}( Q_{\rho /2}(x)) \ge \mu _0( Q_{\rho /2}(x))\ge \rho ,\) 
a contradiction. If there exists \(x\in L\setminus T_0,\)  then for some \( Q_\rho (x)\cap T_0=\emptyset \)  for some \(\rho >0\)  and there exists a sequence \(x_k\in \sigma _{0,r_k}(\Gamma )\)  such that \(x_k\rightarrow x.\)  Then \( Q_{\rho /2}(x_k)\subset Q_\rho (x)\)  for all large \(k\)  so that by (REF ) and (REF ),
\(0=\mu _0(\overline{ Q_\rho (x)}) \ge \limsup \limits _{k\rightarrow \infty } \mu _{r_k}(\overline{ Q_\rho (x)}) \ge \limsup \limits _{k\rightarrow \infty } \mu _{r_k}( Q_{\rho /2}(x_k))\ge c\rho ,\) 
a contradiction. Thus, \(L=T_0.\)  Since the sequence \(r_k\searrow 0\)  is arbitrary, (REF ) follows. Now (REF ) is obvious.
